A Differentiability

Differentiability refers to the property of a mathematical function having a defined derivative at a specific point or across an entire interval. In geometric terms, a differentiable function is "smooth" and lacks sharp corners or breaks, meaning it can be drawn a unique, non-vertical tangent line at every differentiable point on its curve.

The Formal Definition

A function \(f(x)\) is differentiable at a point \(x = a\) if the following limit exists:

\[f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\]

If this limit approaches a specific real number from both the left and the right, the function is differentiable at \(a\), and that number is the slope of the tangent line.

The Relationship Between Differentiability and Continuity

• Differentiability implies continuity: If a function is differentiable at a point \(a\), it is mathematically guaranteed to be continuous at \(a\).
• Continuity does not imply differentiability: A function can be continuous but still fail to be differentiable.

Where Differentiability Fails

A function is not differentiable at a point \(x = a\) if any of the following occur:

• Sharp Corners: The left-hand slope and right-hand slope do not match.
• Cusps: The curve sharply reverses direction, causing the slopes from both sides to approach \(\infty\) or \(-\infty\) with opposite signs. An example is \(f(x) = x^{2/3}\) at \(x = 0\).
• Vertical Tangents: The graph becomes perfectly vertical for an instant, causing the derivative (the slope) to become infinite. An example is \(f(x) = \sqrt[3]{x}\) at \(x = 0\).
• Discontinuities: Any break, jump, hole, or vertical asymptote in the graph automatically means the function cannot be differentiated there, because it lacks a continuous path to measure a rate of change.